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==Hausdorff measure==
{{main|Hausdorff measure}}
The Hasudorff measure is the most-used fractal measure and provides a definition for [[Hausdorff dimension]], which is in turn one of the most frequently used definitions of fractal dimension. Intuitively, the Hausdorff measure can be thought of as covering the set by other sets, and taking the smallest possible measure of the coverings as the they approach zero.
Let <math>(X,\rho)</math> be a [[metric space]]. For any subset <math>U\subset X</math>, let <math>\mathrm{diam}\;U</math> denote its diameter, that is
:<math>\operatorname{diam} U :=\sup\{\rho(x,y):x,y\in U\}, \quad \operatorname{diam} \emptyset:=0</math>
Let <math>S</math> be any subset of <math>X,</math> and <math>\delta>0</math> a real number.
:<math>H^d_\delta(S)=\inf\left \{\sum_{i=1}^\infty (\operatorname{diam} U_i)^d: \bigcup_{i=1}^\infty U_i\supseteq S, \operatorname{diam} U_i<\delta\right \},</math>
where the infimum is over all countable covers of <math>S</math> by sets <math>U_i\subset X</math> satisfying <math> \operatorname{diam} U_i<\delta</math>
When the ''d''-dimensional Hausdorff measure is an integer, <math>H^d(S)</math> is proportional to the [[Lebesgue measure]] for that dimension. Due to this, some definitions of Hausdorff measure include a scaling by the volume of the unit [[N-sphere|''d''-ball]], expressed using [[gamma function|Euler's gamma function]] as
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==Packing measure==
{{further|Packing dimension}}
Just as the packing dimension is in some ways a dual to the Hausdorff dimension, the packing measure is a counterpart to the Hausdorff measure. The packing measure is defined informally as the measure of "packing" a set with open balls, and calculating the measure of those balls. In contrast to Hausdorff measure, which covers the set being measured and can have intersecting covers, the packing measure requires no balls to intersect and becomes smaller than the Hausdorff measure.
Let (''X'', ''d'') be a metric space with a subset ''S'' ⊆ ''X'' and let ''s'' ≥ 0. We take a "pre-measure" of ''S'', defined to be
:<math>P_0^s (S) = \limsup_{\delta \downarrow 0}\left\{ \left. \sum_{i \in I} \mathrm{diam} (B_i)^s \right| \begin{matrix} \{ B_i \}_{i \in I} \text{ is a countable collection} \\ \text{of pairwise disjoint closed balls with} \\ \text{diameters } \leq \delta \text{ and centres in } S \end{matrix} \right\}.</math><ref>https://projecteuclid.org/download/pdf_1/euclid.rae/1214571371</ref>
The [[pre-measure]] is made into a true [[measure (mathematics)|measure]], where the ''s'''''-dimensional packing measure''' of ''S'' is defined to be
:<math>P^s (S) = \inf \left\{ \left. \sum_{j \in J} P_0^s (S_j) \right| S \subseteq \bigcup_{j \in J} S_j, J \text{ countable} \right\},</math>
i.e., the packing measure of ''S'' is the [[infimum]] of the packing pre-measures of countable covers of ''S''.
==References==
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